A pool measuring 10 meters by 20 meters is surrounded by a path of uniform width. If the area of

A pool measuring 10 meters by 20 meters is surrounded by a...

Key Concepts

Quadratic Equation

The expansion and simplification of the equation typically result in a quadratic equation. Solving this equation—often using the quadratic formula—yields the value of the unknown width of the border.

Setting Up Equations

Translating the verbal description of the problem into an algebraic equation involves expressing the combined area as the product of the new length and width, then setting that product equal to the given total area.

Uniform Border

A uniform border adds a constant width to all sides of a shape. Recognizing that the border extends the original dimensions by twice its width is key to calculating the new dimensions and area.

Area of a Rectangle

Understanding that the area of a rectangle is calculated by multiplying its length by its width is essential. This concept is the basis for determining the area of both the original pool and the larger rectangle formed when the path is added.

Transcript

00:01 So in this problem, we're going to use some information given to us about a pool and a walk away surrounding the pool in order to find out the width of this path around it.

00:12 So let's label the width of the path x, since it's the unknown that we're trying to find.

00:18 And it says it's of uniform width.

00:20 So that means that it's x over here, x over here, x over here, x over here.

00:27 And we know that the pool itself is 10 meters by 20 meters.

00:33 And then the other thing we know is that the area of the entire rectangular situation.

00:38 So the pool and the path combined, it's going to be 600 square meters.

00:43 So we can use this information to write and solve an equation for x.

00:48 So let's find the dimensions of this green rectangle here.

00:55 So we know that from here to here, we have 20 meters.

01:03 And on each of the ends, we have x meters.

01:08 So that means that this entire side can be labeled 20 plus 2x.

01:14 And so that means that the top is also going to be 20 plus 2x.

01:19 We can do a really similar thing on this side over here on the right.

01:23 We know that this middle section will be 10, and then we know that we have x on either side.

01:31 So this side will be 10 plus 2x, same as the other side since it's rectangular.

01:39 So now that we have expressions for our two side lengths, we can write an equation for area, because area of a rectangle is base times height, length, times width, whatever you want to call it.

01:51 So we're going to multiply 20 plus 2x and 10 plus 2x, and that will give us area, which we already know is 600.

02:05 So this will set equal to 600.

02:10 All right.

02:10 So now we've done the work of creating an equation in terms of x.

02:14 Now we have to solve this equation for x...

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